My First Notebook MDA2024

Introduction

1. Data Description

1.1. Sample Selected

You can define bullet list or numbered list:

• Pizza
• Pasta
• Cafe
  • Espresso
  • Macchiato
  • Cappuccino
• Vodka
  • Bisont Grass
  • Soplica

1.2. Formula

Here you can define formula

\[ y = \beta_0 + \beta_1X + \epsilon \] the formula can be reported in the text: \(\mu = 1/n \sum X_i\)

1.3. Import Data (CSV) [*]

• Pre-trained Models:
  • https://www.kaggle.com/models
• Open psychology data [raw data]:
  • http://openpsychometrics.org/_rawdata/
• Info about used data [DS-3]:
  • openpsychometrics.org/tests/SD3/
    • Dark Triad Personality Test [data]
    • SD-3 stands for Short Dark Triad

data_SD3 <- read.delim("~/RProjects/2024-Q2-R-2 [MDA2024, exercises]/D1_SD3/data_SD3.csv", stringsAsFactors=TRUE)

[ 1.3. BONUS ]

• Imported:
  • Forbes2000.csv
  • glass.csv

Forbes2000 <- read.csv("~/RProjects/2024-Q2-R-2 [MDA2024, exercises]/D0_Data/Forbes2000.csv", stringsAsFactors=TRUE)

glass <- read.csv("~/RProjects/2024-Q2-R-2 [MDA2024, exercises]/D0_Data/glass.csv", stringsAsFactors=TRUE)

2. Data Analysis

To add R code in the Notebook we need to use the Chunk.

X <- iris

It is possible to have an overview of the data by using the summary function.

summary(X)

  Sepal.Length    Sepal.Width     Petal.Length    Petal.Width          Species  
 Min.   :4.300   Min.   :2.000   Min.   :1.000   Min.   :0.100   setosa    :50  
 1st Qu.:5.100   1st Qu.:2.800   1st Qu.:1.600   1st Qu.:0.300   versicolor:50  
 Median :5.800   Median :3.000   Median :4.350   Median :1.300   virginica :50  
 Mean   :5.843   Mean   :3.057   Mean   :3.758   Mean   :1.199                  
 3rd Qu.:6.400   3rd Qu.:3.300   3rd Qu.:5.100   3rd Qu.:1.800                  
 Max.   :7.900   Max.   :4.400   Max.   :6.900   Max.   :2.500                  

In R there are three main type of data:

• Matrix. Mathematical Object. In our example Y is a matrix.
• Data Frame. It is devoted to organize and analyze data and it is a generalization of the Matrix. In our example, X is a data frame.
• List. It is an object that can include Data Frames, Matrices or other lists.

Y <- as.matrix(X[ ,1:4])

To handle data you can use the following code:

X[10, 2]       # selection of one element in the Data Frame (or matrix)

[1] 3.1

X[5:20, 1:3]   # selection of an interval

X[5:20, ]      # the empty space select all the columns or rows

X$Sepal.Length # the symbol $ is used to select a column in the data frame

  [1] 5.1 4.9 4.7 4.6 5.0 5.4 4.6 5.0 4.4 4.9 5.4 4.8 4.8 4.3 5.8 5.7 5.4 5.1 5.7 5.1 5.4 5.1
 [23] 4.6 5.1 4.8 5.0 5.0 5.2 5.2 4.7 4.8 5.4 5.2 5.5 4.9 5.0 5.5 4.9 4.4 5.1 5.0 4.5 4.4 5.0
 [45] 5.1 4.8 5.1 4.6 5.3 5.0 7.0 6.4 6.9 5.5 6.5 5.7 6.3 4.9 6.6 5.2 5.0 5.9 6.0 6.1 5.6 6.7
 [67] 5.6 5.8 6.2 5.6 5.9 6.1 6.3 6.1 6.4 6.6 6.8 6.7 6.0 5.7 5.5 5.5 5.8 6.0 5.4 6.0 6.7 6.3
 [89] 5.6 5.5 5.5 6.1 5.8 5.0 5.6 5.7 5.7 6.2 5.1 5.7 6.3 5.8 7.1 6.3 6.5 7.6 4.9 7.3 6.7 7.2
[111] 6.5 6.4 6.8 5.7 5.8 6.4 6.5 7.7 7.7 6.0 6.9 5.6 7.7 6.3 6.7 7.2 6.2 6.1 6.4 7.2 7.4 7.9
[133] 6.4 6.3 6.1 7.7 6.3 6.4 6.0 6.9 6.7 6.9 5.8 6.8 6.7 6.7 6.3 6.5 6.2 5.9

2.1. Plots in R

# boxplot - horizontal True (T) [ with an error ]
# error type: Error in if (horizontal) plot.window(ylim = xlim, xlim = ylim, log = log,  :  the condition has length > 1
# boxplot(X$Sepal.Length, main = "Box Plot of the Sepal Length of the IRIS Flowers", col = "blue", horizontal = T)
# boxplot - horizontal False (F) [ without error ]
boxplot(X$Sepal.Length, main = "Box Plot of the Sepal Length of the IRIS Flowers", col = "blue", horizontal = F)

The elements of the box-plot are reported below:

• Q1. It is the First Quartile. It leaves 25% of units on the left and 75% of units on the right. Left side of the box
• Me. It is the Median=Q2. It leaves 50% of units on left and right. Bold Line in the middle of the box
• Q3. It is the Third Quartile. It leaves 75% of the units on the left and 25% of the units on the right. Right side of the box
• In case of no outliers, the whiskers are defined as:
  • Xmin is the left whiskers
  • Xmax is the right whiskers
• In case of outliers, the whiskers are defined as:
  • Linf = Q1 - 1.5 (Q3-Q1): Lower Limit
  • Lsup = Q1 + 1.5 (Q3-Q1): Upper Limit

# just a boxplot
boxplot(X[ ,1:4])

# boxplot with a title and colors
boxplot(X[ ,1:4], main = "Box Plot of all quantitative variables of IRIS data", col = terrain.colors(4))

boxplot(X$Sepal.Width ~ X$Species, main = "Box Plot of Sepal Width considering the 3 types of flowers", xlab = "Type of IRIS Flowers", ylab = "Sepal Width", col = terrain.colors(4))

The tilde symbol is obtained by:

• these doesn’t work [*]:
  • MAC: option + 5
  • WIN: ALT + 125/6
• these works [*]:
  • MAC: shift + button before no.1
  • WIN: shift + button before no.1

[ 2.1. BONUS ]

The bold line is the Median, that is the value of the ordered distribution that leaves the same number of units above and below (or on left and right)

• Q1 is the first quartile. Q1 leaves 25% of unirs on left and 75% on right.
• Q3 is the third quartile. Q3 leaves 75% of units on left and 25% on right.
• Wishers, without outliers are, the min and max of distribution
• Wishers, with outliers are, Lmin = Q1-1.5(Q3-Q1); Lsup = Q3+1.5(Q3-Q1);

boxplot(X$Sepal.Length, main = "Box-Plot of the Sepal Lenght", col = "green", horizontal = F)

boxplot(X[ ,1:4], main = "Box-Plot with all the Variables", col = "blue", horizontal = F)

boxplot(X$Sepal.Width ~ X$Species, main = "Box-Plot about Sepal Width with different type of IRIS Flowers")

2.2. Bar Plot

Bar Plot can be used for Qualitative Data and for Categorized Quantitative Data. The first step to create a Bar Plot is to generate a Table of Frequency.

T <- table(X$Species)
T

    setosa versicolor  virginica 
        50         50         50 

barplot(T, main = "Bar Plot of Type of flowers", xlab = "Type of flowers", ylab = "Absolute Frequency", col = terrain.colors(4))

2.3. Pie Chart

It is based on the frequency table.

pie(T, main = "Pie Chart", col = terrain.colors(4))

2.4. Histogram Chart

Histogram is a plot used only for Quantitative Data, it is based on a frequency tables in classes. The R function is called hist and the input is a simple distribution of a quantitative variable.

hist(X$Sepal.Length)

hist(X$Sepal.Width, main = "Histogram of Sepal Width", xlab = "Classes", ylab = "Absolute Frequency", col = "lightgreen", border = "blue")

[ 2.4. BONUS ]

he histogram can be used only for quantitative variables.

hist(X$Sepal.Width, main = "Histogram of the Sepal Width", xlab = "Classes",
     ylab = "Absolute Frequency", col = "green", border = "red", breaks = 10)

In case of equally spaced (same size) classes we can report on the Y axis the Absolute Frequency or relative frequency. In case of classes with different sizes we have to report on Y axis the density of frequency. The formula is the following: \(d_i = n_i/h_i\), where \(n_i\) is the absolute frequency and \(h_i\) is the size of the class.

3. Correlation Analysis

3.1. Correlation Plot

plot(X$Sepal.Length, X$Sepal.Width, main = "Correlation Plot", xlab = "Sepal Length", ylab = "Sepal Width")

[ 3.1. BONUS ]

plot(X$Petal.Length, X$Petal.Width, main = "Correlation Plot",
     xlab = "Petal Lenght", ylab = "Petal Width", col = "blue",
     pch = 1)

Plots with IRIS

plot(X$Sepal.Length, X$Sepal.Width, main = "(1-2) Plot with IRIS", 
     xlab = "Sepal Lenght", ylab = "Sepal Width", col = "blue")

plot(X$Petal.Length, X$Petal.Width, main = "(2-2) Plot with IRIS", 
     xlab = "Petal Lenght", ylab = "Petal Width", col = "red")

3.2. Pair Plot

pairs(X[ ,1:4])

The plots below the main diagonal are the same of the plot above the main diagonal. The reason is because the plot and the correlation index are symmetric.

r <- cor(X[ ,1:2])
r <- round(r, 3)
r

             Sepal.Length Sepal.Width
Sepal.Length        1.000      -0.118
Sepal.Width        -0.118       1.000

cor(X$Sepal.Length, X$Sepal.Width)

[1] -0.1175698

The range of correlation index is: -1 <= r <= 1 The interpretation of the Correlation Index called r is following:

• 0.00 < |r| <= 0.25 Low Correlation
• 0.25 < |r| <= 0.50 Medium-Low Correlation
• 0.50 < |r| <= 0.75 Medium-High Correlation
• 0.75 < |r| <= 1.00 High Correlation
• 0 No Correlation
• 1 Perfect Correlation

The correlation between Sepal Length and Sepal Width is -0.118 and it is a low negative correlation.

4. Plots with GGPLOT Package

install.packages("ggplot2")

Error in install.packages : Updating loaded packages

library(ggplot2)

GGPLOTS has 3 main arguments:

• The first argument is the data: ggplot(data = X). It creates an empty frame.
• The second argument is the geometry (type of plot): geom_. It adds a layer with the type of plot we want to show.
• The third argument is the aesthetic, to select the variables and the properties: mapping = aes()

# GGPLOT (example no 1)
ggplot(data = X[ ,1:2])

# GGPLOT (example no 2.1)
ggplot(data = X[ ,1:2]) +
  geom_point(mapping = aes(X$Sepal.Length, X$Sepal.Width))

# GGPLOT (example no 2.2)
ggplot(data = X[ ,1:2]) +
  geom_point(mapping = aes(Sepal.Length, Sepal.Width))

# GGPLOT (example no 2.2)
ggplot(data = X) +
  geom_point(mapping = aes(Sepal.Length, Sepal.Width))

# GGPLOT (example no 3)
ggplot(data = X) +
  geom_point(mapping = aes(Sepal.Length, Sepal.Width)) +
  ggtitle("Scatter Plot") + xlab("Sepal Length") + ylab("Sepal Width")

4.1. Correlation Plot (Scatter Plot) with colors

ggplot(data = X) +
  geom_point(mapping = aes(Sepal.Length, Sepal.Width, color = Species))

[ 4.1. BONUS ]

STANDARD TEMPLATE IS: ggplot(data = ) + (mapping = aes())

# First Example
ggplot(data = X) + 
  geom_point(mapping = aes(Petal.Length, Petal.Width, color = Species)) +
  ggtitle("Petal Lenght and Width") +
  xlab("Petal Lenght") + ylab("Petal Width")

# Second Example
ggplot(data = X, mapping = aes(Petal.Width, Petal.Length)) + 
  geom_point(mapping = aes(color = Species)) +
  ggtitle("Petal Width and Lenght") + 
  xlab("Petal Width") + ylab("Petal Lenght")

4.2. Box Plot

ggplot(data = X) +
  geom_boxplot(mapping = aes(Sepal.Width), color = "blue", outlier.colour = "red", outlier.shape = 8, outlier.size = 3) +
  ggtitle("Box Plot for Sepal Lenght")

Box Plot taking into account the 3 types of flowers

ggplot(data = X) +
  geom_boxplot(mapping = aes(Species, Sepal.Width), outlier.color = "red", outlier.shape = 8)

GGPLOT function as object

p <- ggplot(data = X) +
  geom_boxplot(mapping = aes(Species, Sepal.Width, fill = Species))
p

p + theme(legend.position = "bottom")

4.3. Bar Plot

ggplot(data = X) + 
  geom_bar(mapping = aes(Species)) + 
  ggtitle("Bar Plot with GGPLOT") + 
  ylab("Absolute Frequency")

5. Use the mpg data (mpg Fuel economy data from 1999 to 2008 for 38 popular model of cars)

A data frame with 234 rows and 11 variables:

• manufacturer brand name
• model model name
• displ engine displacement, in litres (power of the engine)
• year year of manufacture
• cyl number of cylinders
• trans type of transmission
• drv the type of drive train, where f = front-wheel drive, r = rear wheel drive, 4 = 4wd
• cty city miles per gallon (km per liter in town)
• hwy highway miles per gallon (km per liter in highway)
• fl fuel type
• class “type” of car

Y <- mpg

summary(Y)

 manufacturer          model               displ            year           cyl       
 Length:234         Length:234         Min.   :1.600   Min.   :1999   Min.   :4.000  
 Class :character   Class :character   1st Qu.:2.400   1st Qu.:1999   1st Qu.:4.000  
 Mode  :character   Mode  :character   Median :3.300   Median :2004   Median :6.000  
                                       Mean   :3.472   Mean   :2004   Mean   :5.889  
                                       3rd Qu.:4.600   3rd Qu.:2008   3rd Qu.:8.000  
                                       Max.   :7.000   Max.   :2008   Max.   :8.000  
    trans               drv                 cty             hwy             fl           
 Length:234         Length:234         Min.   : 9.00   Min.   :12.00   Length:234        
 Class :character   Class :character   1st Qu.:14.00   1st Qu.:18.00   Class :character  
 Mode  :character   Mode  :character   Median :17.00   Median :24.00   Mode  :character  
                                       Mean   :16.86   Mean   :23.44                     
                                       3rd Qu.:19.00   3rd Qu.:27.00                     
                                       Max.   :35.00   Max.   :44.00                     
    class          
 Length:234        
 Class :character  
 Mode  :character  
                   
                   
                   

head(Y)

5.1. Bar Chart

table(Y$cyl)

 4  5  6  8 
81  4 79 70 

# first example: factor(cyl) as a color ( vertical bar chart )
ggplot(data = Y) +
  geom_bar(mapping = aes(cyl, fill = factor(cyl)))

# second example: class as a color ( vertical bar chart )
ggplot(data = Y) +
  geom_bar(mapping = aes(cyl, fill = class))

# third example: class instead cyl ( vertical bar chart )
ggplot(data = Y) +
  geom_bar(mapping = aes(cyl, fill = class))

# fourth example: class instead cyl ( horizontal bar chart )
ggplot(data = Y) +
  geom_bar(mapping = aes(cyl, fill = class)) + coord_flip()

# fifth example: class instead cyl ( horizontal bar chart & legend at the bottom )
ggplot(data = Y) +
  geom_bar(mapping = aes(cyl, fill = class)) + coord_flip() + theme(legend.position = "bottom")

5.2. Histogram

# example no 1.1
ggplot(data = Y) +
  geom_histogram(mapping = aes(cty))

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# example no 1.2
ggplot(data = Y) +
  geom_histogram(mapping = aes(cty, colour = class))

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# example no 1.3
ggplot(data = Y) +
  geom_histogram(mapping = aes(cty, fill = class))

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# example no 1.4
ggplot(data = Y) +
  geom_histogram(mapping = aes(cty, fill = "red"))

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# example no 1.5
ggplot(data = Y) +
  geom_histogram(mapping = aes(cty, fill = factor(cty)))

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# example 2
ggplot(data = Y) +
  geom_histogram(mapping = aes(hwy))

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

5.3. Facet Wrap with 1 grup variable [*]

With Facet Wrap layer (option) we can create sub-plots based on a categorical variable.

ggplot(data = Y) +
  geom_point(mapping = aes(displ, hwy)) +
  facet_wrap(~drv)

5.4. Facet Wrap with 2 grup variables [*]

ggplot(data = Y) +
  geom_point(mapping = aes(displ, hwy)) +
  facet_wrap(drv ~ class)

5.5. Smooting Plot [*]

ggplot(data = Y) +
  geom_smooth(mapping = aes(displ, hwy))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

5.5.1. Smooting Plot with “different type of line” [*]

# First Example
ggplot(data = Y) +
  geom_smooth(mapping = aes(displ, hwy, linetype = class))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  span too small.   fewer data values than degrees of freedom.
Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  pseudoinverse used at 5.6935
Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  neighborhood radius 0.5065
Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  reciprocal condition number  0
Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  There are other near singularities as well. 0.65044
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  span too small.   fewer data values than degrees of freedom.
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  pseudoinverse used at 5.6935
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  neighborhood radius 0.5065
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  reciprocal condition number  0
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  There are other near singularities as well. 0.65044
Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  pseudoinverse used at 4.008
Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  neighborhood radius 0.708
Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  reciprocal condition number  1.6135e-17
Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,  :
  There are other near singularities as well. 0.25
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  pseudoinverse used at 4.008
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  neighborhood radius 0.708
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  reciprocal condition number  1.6135e-17
Warning in predLoess(object$y, object$x, newx = if (is.null(newdata)) object$x else if (is.data.frame(newdata)) as.matrix(model.frame(delete.response(terms(object)),  :
  There are other near singularities as well. 0.25

# Second Example
ggplot(data = Y) +
  geom_smooth(mapping = aes(displ, hwy, linetype = drv))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

5.5.2. Smooting Plot with “Facet Wrap” [*]

ggplot(data = Y) +
  geom_smooth(mapping = aes(displ, hwy)) +
  facet_wrap(~ drv)

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

5.5.3. Smooting Plot with “color” [*]

ggplot(data = Y) +
  geom_smooth(mapping = aes(displ, hwy, color = drv))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

5.5.4. Smooting Plot with “group” [*]

ggplot(data = Y) +
  geom_smooth(mapping = aes(displ, hwy, group = drv))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

5.5.5. Smooting Plot combining different layers [*]

ggplot(data = Y) +
  geom_point(mapping = aes(displ, hwy, color = drv)) +
  geom_smooth(mapping = aes(displ, hwy, color = drv))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

6. Regression Model [*]

In regression model we need to define the dependent and independent variables. In our case the model is define as follow:

• Y(Dependent/Outcome Variable) = hwy. The variable defines the number of miles (km) per Gallon (liter) on the highway.
• X(Independent/Input Variable) = displ. The variable defines the power of the engine (horse power).

In the first place we need to create a scatter plot (correlation plot).

6.1. Regression Model: Plot [*]

ggplot(data = Y) +
  geom_point(mapping = aes(displ, hwy)) +
  geom_smooth(method = lm, mapping = aes(displ, hwy))

`geom_smooth()` using formula = 'y ~ x'

6.2. Regression Model: Different Plot per each group [*]

ggplot(data = Y) +
  geom_point(mapping = aes(displ, hwy, color  = drv)) +
  geom_smooth(method = lm, mapping = aes(displ, hwy, color = drv))

`geom_smooth()` using formula = 'y ~ x'

6.3. Regression Model: Parameters Estimation [*]

res.reg <- lm(hwy ~ displ, data = Y)
summary(res.reg)

Call:
lm(formula = hwy ~ displ, data = Y)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.1039 -2.1646 -0.2242  2.0589 15.0105 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  35.6977     0.7204   49.55   <2e-16 ***
displ        -3.5306     0.1945  -18.15   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.836 on 232 degrees of freedom
Multiple R-squared:  0.5868,    Adjusted R-squared:  0.585 
F-statistic: 329.5 on 1 and 232 DF,  p-value: < 2.2e-16

The regression coefficients are defined as follow:

• \(\beta_0\) is the intercept and is the value of Y when X is equal to zero
• \(\beta_1\) is the regression coefficient and is the slope of the line. It is the value of Y when there is a variant of X equal to 1: \(\Delta_x=1\)

6.3.1. Regression Parameters

The results of the regression model showed that the estimated \(\beta_0\) (the intercept) is equal to 35.6977 and the estimated \(\beta_1\) (the regression coefficient/slope) is equal to -3.3506.

6.3.2. Godness of Fit

The quality of the model is measured by the godness of fix index \(R^2\) that is for this case study equal to 0.585. The \(R^2\) is interpreted as follow:

• 0.00 < \(R^2\) <= 0.25 Low Quality
• 0.25 < \(R^2\) <= 0.50 Medium-Low Quality
• 0.50 < \(R^2\) <= 0.75 Medium-High Quality
• 0.75 < \(R^2\) < 1.00 High Quality
• \(R^2\) = 0 Non connection between Y and X. The result will be on horizontal line
• \(R^2\) = 1 Perfect connection between Y and X. The data points will be exactly on the regression line

In this case study we have a Medium-High Quality of the model.

6.3.3. Test on the estimated Regression Parameters

To test the regression parameters it is possible to use the t-value or the p-value, as reported below:

• Approach by using the t-value:
  • -2 < t-value < 2 we accept the null hypothesis (BAD). The relation between Y and X is not statistically significant.
  • t-value > 2 we reject the null hypothesis (GOOD). The relation between Y and X is statistically significant
  • t-value < -2 we reject the null hypothesis (GOOD). The relation between Y and X is statistically significant.
• Approach by using the p-value:
  • p-value > 0.05 (5%) we accept the null hypothesis (BAD). The relation between Y and X is not statistically significant.
  • p-value < 0.05 (5%) we reject the null hypothesis (GOOD). The relation between Y and X is statistically significant.

In this case study the estimated regression coefficient \(\beta_1\) is statistically significant, since the t-value is -18.15 (< 2) and the p-value is 2e-16 (< 5%). It means that the number of mile per gallon (hwy) depends by the power of the engine (displ), so by increasing the displ of 1 unit, the hwy will decrease by -3.5306.

7. Principal Component Analysis (PCA)

library(FactoMineR)
library(factoextra)
library(ggplot2)
AUTO <- read.csv("~/RProjects/2024-Q2-R-2 [MDA2024, exercises]/D2_AUTO/AUTO.csv", row.names=1, dec=",")

res.pca <- PCA(AUTO, graph = F)

7.1. Summary of the Results

summary.PCA(res.pca)

Call:
PCA(X = AUTO, graph = F) 


Eigenvalues
                       Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6   Dim.7   Dim.8   Dim.9
Variance               3.184   2.032   1.489   1.339   0.985   0.400   0.335   0.132   0.085
% of var.             31.844  20.321  14.892  13.389   9.852   4.000   3.348   1.319   0.849
Cumulative % of var.  31.844  52.166  67.058  80.447  90.299  94.299  97.648  98.967  99.816
                      Dim.10
Variance               0.018
% of var.              0.184
Cumulative % of var. 100.000

Individuals (the 10 first)
                 Dist    Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2  
A155         |  4.094 |  3.053 19.520  0.556 | -2.037 13.619  0.248 | -0.321  0.461  0.006 |
AU80         |  3.722 | -1.451  4.406  0.152 | -1.774 10.321  0.227 |  2.480 27.541  0.444 |
BMW3         |  4.306 |  3.181 21.190  0.546 | -0.561  1.031  0.017 |  0.016  0.001  0.000 |
CXAN         |  3.093 |  1.072  2.405  0.120 |  2.357 18.229  0.581 | -0.602  1.622  0.038 |
TEMP         |  1.642 | -0.056  0.007  0.001 | -0.708  1.644  0.186 |  1.190  6.336  0.525 |
MOND         |  3.695 |  2.894 17.536  0.614 |  1.762 10.182  0.227 |  0.327  0.477  0.008 |
DELT         |  3.818 |  0.085  0.015  0.000 | -1.886 11.669  0.244 |  0.601  1.615  0.025 |
DEDR         |  1.375 |  0.151  0.048  0.012 | -0.798  2.087  0.337 | -0.248  0.275  0.032 |
PRIM         |  3.431 | -1.195  2.990  0.121 | -0.989  3.208  0.083 | -2.298 23.646  0.449 |
VECT         |  2.335 | -0.808  1.366  0.120 |  1.087  3.874  0.217 | -0.303  0.411  0.017 |

Variables
                Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2  
prezzo       |  0.445  6.205  0.198 | -0.087  0.372  0.008 |  0.379  9.665  0.144 |
cilindrata   |  0.659 13.642  0.434 |  0.403  8.009  0.163 |  0.058  0.223  0.003 |
cavalli      |  0.849 22.634  0.721 | -0.399  7.853  0.160 |  0.018  0.022  0.000 |
lunghezza    | -0.111  0.387  0.012 |  0.542 14.444  0.294 |  0.213  3.046  0.045 |
larghezza    |  0.334  3.513  0.112 |  0.742 27.064  0.550 |  0.177  2.113  0.031 |
peso         |  0.652 13.357  0.425 | -0.333  5.450  0.111 |  0.284  5.400  0.080 |
velocita     |  0.923 26.770  0.852 |  0.001  0.000  0.000 |  0.144  1.389  0.021 |
cons_strada  | -0.191  1.149  0.037 | -0.805 31.881  0.648 |  0.425 12.147  0.181 |
cons_urbano  | -0.091  0.261  0.008 |  0.315  4.898  0.100 |  0.709 33.791  0.503 |
affidabilita | -0.620 12.079  0.385 | -0.025  0.030  0.001 |  0.693 32.205  0.480 |

7.2. Analyze the Eigen Values

The aim of PCA is to create new “Latent Variable” (Factominer language = “Comp” - Components) as linear combination of the original ones. The number of LVs should be equal to q < than p, where “p” is the number of the manifest/original variables, in this cases study p=10. To select the number q of LVs we use the eigenvalues that represents the level of information extracted (explained variance extracted) from the original data. The rules to select the number of LVs are the following: - Eigenvalues greater than 1: \(\lambda > 1\) - Cumulative percentage of explained variance \(> 70%\) (\(>0.7\)) (around \(70%\))

In this case, considering the two eigenvalues criteria, we should select for the \(\lambda > 1\) four LVs; for the second criteria we select only 3 LVs. In general, for the exame report, you select max 2 LVs.

round(res.pca$eig, 3)

        eigenvalue percentage of variance cumulative percentage of variance
comp 1       3.184                 31.844                            31.844
comp 2       2.032                 20.321                            52.166
comp 3       1.489                 14.892                            67.058
comp 4       1.339                 13.389                            80.447
comp 5       0.985                  9.852                            90.299
comp 6       0.400                  4.000                            94.299
comp 7       0.335                  3.348                            97.648
comp 8       0.132                  1.319                            98.967
comp 9       0.085                  0.849                            99.816
comp 10      0.018                  0.184                           100.000

7.3. Variables Plot (Variables Dimension - Circle of Correlation)

The table reported below is called table of correlation or table of the coordinates, since in case the data are standardized, correlations and coordinations have the same values. The table reports the correlations between the Manifest Variables (original variables) and the LVs (Dimensions in Factominer language). The correlations represent also the coordinates of the manifest on the Variable Plot.

For the interpretation:

• Higher is the correlation (biggher is coordinate), higher is important of a MV for a specific LV (Dimension). In this case study, for the first LV (Dimension) the most important variable is velocita (speed), since the correlation with Dim.1 is 0.923. For the second LV (Dim.2) is cons_strada (km per liter in Highway), since the correlation is -0.805. The correlation have to be read in absolute way
• The sign of the correlation give information about the position of the manifest variables
  • On the first dimension (x-axis): Positive Sign, the MV is on the right site; Negative Sign, the MV is on the left side.
  • On the second dimension (y-axis): Positive Sign, the MV is on the upper side; Negative Sign, the MV is on the lower side [on the plot].
• We can give a meaning (interpretation) to the Latent Variables (Dimensions), based on correlation table, that means on the position of the MVs on the Variables Plot.

In this case study the two LVs (Dimensions) are so defined: - LV_1-Dim.1: Right Side: Cilindrata, Cavalli, Velocita; Left Side: Affidabilita (reliability) - LV_2-Dim.2: Upper Side: Lunghezza, Larghezza (lenght, large); Lower Side: cons_strada (km per liter)

round(res.pca$var$coord, 3)

              Dim.1  Dim.2 Dim.3  Dim.4  Dim.5
prezzo        0.445 -0.087 0.379  0.434 -0.608
cilindrata    0.659  0.403 0.058 -0.389 -0.333
cavalli       0.849 -0.399 0.018  0.162  0.172
lunghezza    -0.111  0.542 0.213  0.743  0.152
larghezza     0.334  0.742 0.177 -0.068  0.486
peso          0.652 -0.333 0.284  0.110  0.418
velocita      0.923  0.001 0.144  0.012 -0.077
cons_strada  -0.191 -0.805 0.425 -0.162  0.178
cons_urbano  -0.091  0.315 0.709 -0.574 -0.035
affidabilita -0.620 -0.025 0.693  0.219 -0.044

plot.PCA(res.pca, choix = "var")

It is possible to evaluate the correlation between MVs, by using the angles between the vectors (arrows):

• When the angle is 90 degree (about) there is no correlation. For instance between larghezza and cavalli
• When the angle is < 90, there is positive correlation. For instance between larghezza and lunghezza
• When the angle is > 90, there is negative correlation. For instance between velocita and affidabilita

The table reported below is the *Table of \(\cos^2\)** that gives information about the quality of the graphical representation about the MVs. The interpretation is made as follow:

• 0.00 < \(\cos^2\) <= 0.25: Low Quality
• 0.25 < \(\cos^2\) <= 0.50: Medium-Low Quality
• 0.50 < \(\cos^2\) <= 0.75: Medium-High Quality
• 0.75 < \(\cos^2\) <= 1.00: High Quality

For instance, in the case study, cavalli has Medium-High Quality (\(\cos^2 = 0.721\)) and velocita has High Quality since \(\cos^2 = 0.825\).

round(res.pca$var$cos2, 3)

             Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
prezzo       0.198 0.008 0.144 0.189 0.370
cilindrata   0.434 0.163 0.003 0.152 0.111
cavalli      0.721 0.160 0.000 0.026 0.030
lunghezza    0.012 0.294 0.045 0.551 0.023
larghezza    0.112 0.550 0.031 0.005 0.236
peso         0.425 0.111 0.080 0.012 0.175
velocita     0.852 0.000 0.021 0.000 0.006
cons_strada  0.037 0.648 0.181 0.026 0.032
cons_urbano  0.008 0.100 0.503 0.330 0.001
affidabilita 0.385 0.001 0.480 0.048 0.002

7.4. Units Plot

The table reported below is called Table of the Units Coordinates and gives information about the position of the units, in this case study about the cars, on the plot, based on the meaning (interpretation) given by the Variable Plot. In the Units Plot, the coordinates are not the correlations.

round(res.pca$ind$coord, 3)

      Dim.1  Dim.2  Dim.3  Dim.4  Dim.5
A155  3.053 -2.037 -0.321  0.467  1.120
AU80 -1.451 -1.774  2.480  1.220  0.532
BMW3  3.181 -0.561  0.016  1.604 -2.214
CXAN  1.072  2.357 -0.602 -0.539  1.164
TEMP -0.056 -0.708  1.190 -0.572 -0.384
MOND  2.894  1.762  0.327 -0.238  1.164
DELT  0.085 -1.886  0.601 -3.228  0.028
DEDR  0.151 -0.798 -0.248 -0.279  0.102
PRIM -1.195 -0.989 -2.298  1.561  1.247
VECT -0.808  1.087 -0.303 -0.521 -1.359
P405 -3.257 -0.803 -0.896  0.395  0.143
RE21 -1.569  1.885  1.533  0.358  0.371
GOLF -0.707  0.484 -1.798 -1.053 -0.527
PASS -0.836  1.685  1.151  0.722 -0.114
VOL4 -0.559  0.297 -0.832  0.103 -1.272

plot.PCA(res.pca, choix = "ind")

7.5. [BONUS]

The Principal Component Analysis has the following aims and characteristics:

• PCA is only for Quantitative DATA
• Data Reduction by creating a new set of Latent Variables in a number q < p, where p is the the numer of the manifest/original variables.
• Discover Latent Traits/Latent Variables in the data.

library(FactoMineR)
library(ggplot2)
library(factoextra)

For the PCA we will use the PCA function in the Package FactoMineR.

Y <- mpg

7.5.1 PCA for mpg Data

res.pca <- PCA(Y[,c(3,5,8,9)])

summary(res.pca)

Call:
PCA(X = Y[, c(3, 5, 8, 9)]) 


Eigenvalues
                       Dim.1   Dim.2   Dim.3   Dim.4
Variance               3.509   0.379   0.071   0.041
% of var.             87.736   9.477   1.772   1.014
Cumulative % of var.  87.736  97.214  98.986 100.000

Individuals (the 10 first)
          Dist    Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2  
1     |  2.002 |  1.832  0.408  0.837 | -0.600  0.406  0.090 | -0.187  0.212  0.009 |
2     |  2.211 |  2.190  0.584  0.981 | -0.289  0.094  0.017 | -0.087  0.046  0.002 |
3     |  2.202 |  2.160  0.568  0.963 | -0.129  0.019  0.003 | -0.056  0.019  0.001 |
4     |  2.203 |  2.196  0.587  0.994 | -0.118  0.016  0.003 |  0.000  0.000  0.000 |
5     |  0.709 |  0.336  0.014  0.225 | -0.077  0.007  0.012 | -0.498  1.493  0.493 |
6     |  0.731 |  0.575  0.040  0.619 |  0.130  0.019  0.032 | -0.431  1.119  0.348 |
7     |  0.720 |  0.543  0.036  0.568 |  0.340  0.130  0.222 | -0.290  0.506  0.162 |
8     |  1.822 |  1.581  0.304  0.753 | -0.879  0.872  0.233 | -0.122  0.089  0.004 |
9     |  1.781 |  1.258  0.193  0.499 | -1.180  1.569  0.439 | -0.167  0.167  0.009 |
10    |  1.954 |  1.910  0.444  0.955 | -0.408  0.188  0.044 |  0.010  0.001  0.000 |

Variables
         Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2  
displ | -0.932 24.773  0.869 |  0.309 25.155  0.095 |  0.187 49.213  0.035 |
cyl   | -0.933 24.821  0.871 |  0.307 24.860  0.094 | -0.183 47.082  0.033 |
cty   |  0.951 25.754  0.904 |  0.271 19.334  0.073 |  0.038  2.010  0.001 |
hwy   |  0.930 24.652  0.865 |  0.341 30.652  0.116 | -0.035  1.694  0.001 |

7.5.2 PCA on mtcars

Z <- mtcars

res.pca1 <- PCA(Z)

Warning: ggrepel: 10 unlabeled data points (too many overlaps). Consider increasing max.overlaps

summary(res.pca1)

Call:
PCA(X = Z) 


Eigenvalues
                       Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6   Dim.7   Dim.8   Dim.9
Variance               6.608   2.650   0.627   0.270   0.223   0.212   0.135   0.123   0.077
% of var.             60.076  24.095   5.702   2.451   2.031   1.924   1.230   1.117   0.700
Cumulative % of var.  60.076  84.172  89.873  92.324  94.356  96.279  97.509  98.626  99.327
                      Dim.10  Dim.11
Variance               0.052   0.022
% of var.              0.473   0.200
Cumulative % of var.  99.800 100.000

Individuals (the 10 first)
                      Dist    Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2
Mazda RX4         |  2.234 | -0.657  0.204  0.087 |  1.735  3.551  0.604 | -0.601  1.801  0.072
Mazda RX4 Wag     |  2.081 | -0.629  0.187  0.091 |  1.550  2.833  0.555 | -0.382  0.728  0.034
Datsun 710        |  2.987 | -2.779  3.653  0.866 | -0.146  0.025  0.002 | -0.241  0.290  0.007
Hornet 4 Drive    |  2.521 | -0.312  0.046  0.015 | -2.363  6.584  0.879 | -0.136  0.092  0.003
Hornet Sportabout |  2.456 |  1.974  1.844  0.646 | -0.754  0.671  0.094 | -1.134  6.412  0.213
Valiant           |  3.014 | -0.056  0.001  0.000 | -2.786  9.151  0.855 |  0.164  0.134  0.003
Duster 360        |  3.187 |  3.003  4.264  0.888 |  0.335  0.132  0.011 | -0.363  0.656  0.013
Merc 240D         |  2.841 | -2.055  1.998  0.523 | -1.465  2.531  0.266 |  0.944  4.439  0.110
Merc 230          |  3.733 | -2.287  2.474  0.375 | -1.984  4.639  0.282 |  1.797 16.094  0.232
Merc 280          |  1.907 | -0.526  0.131  0.076 | -0.162  0.031  0.007 |  1.493 11.103  0.613
                   
Mazda RX4         |
Mazda RX4 Wag     |
Datsun 710        |
Hornet 4 Drive    |
Hornet Sportabout |
Valiant           |
Duster 360        |
Merc 240D         |
Merc 230          |
Merc 280          |

Variables (the 10 first)
                     Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2  
mpg               | -0.932 13.143  0.869 |  0.026  0.026  0.001 | -0.179  5.096  0.032 |
cyl               |  0.961 13.981  0.924 |  0.071  0.191  0.005 | -0.139  3.073  0.019 |
disp              |  0.946 13.556  0.896 | -0.080  0.243  0.006 | -0.049  0.378  0.002 |
hp                |  0.848 10.894  0.720 |  0.405  6.189  0.164 |  0.111  1.960  0.012 |
drat              | -0.756  8.653  0.572 |  0.447  7.546  0.200 |  0.128  2.598  0.016 |
wt                |  0.890 11.979  0.792 | -0.233  2.046  0.054 |  0.271 11.684  0.073 |
qsec              | -0.515  4.018  0.266 | -0.754 21.472  0.569 |  0.319 16.255  0.102 |
vs                | -0.788  9.395  0.621 | -0.377  5.366  0.142 |  0.340 18.388  0.115 |
am                | -0.604  5.520  0.365 |  0.699 18.440  0.489 | -0.163  4.234  0.027 |
gear              | -0.532  4.281  0.283 |  0.753 21.377  0.567 |  0.229  8.397  0.053 |

7.5.2.1 Eigenvalues interpretation

The eigenvalues measure the level of information retained by the PCA. They show the explained variance from the data and help us to select the number of Latent Variables (or Latent Traits, Principal Components, Dimensions). The rules to select the number of LVs are: - Eigenvalues greater than 1. \(\lambda > 1\). - Cumulative percentage of explained variance > 0.7 (70%)

In our case study we select two components. Both components have eigenvalue greater than 1 and with and Cumulative percentage of explained variance equal to 84.172%.

round(res.pca1$eig,3)

        eigenvalue percentage of variance cumulative percentage of variance
comp 1       6.608                 60.076                            60.076
comp 2       2.650                 24.095                            84.172
comp 3       0.627                  5.702                            89.873
comp 4       0.270                  2.451                            92.324
comp 5       0.223                  2.031                            94.356
comp 6       0.212                  1.924                            96.279
comp 7       0.135                  1.230                            97.509
comp 8       0.123                  1.117                            98.626
comp 9       0.077                  0.700                            99.327
comp 10      0.052                  0.473                            99.800
comp 11      0.022                  0.200                           100.000

7.5.2.2 Variables Plot (Variables Dimension)

Below we can read the table with correlations between the Manifest Variables (MVs) and the Latent Variables (Dimensions). The correlations are also the coordinates of the manifest variables on the variables plot.

For the interpretation:

• Higher is the correlation(bigger is the coordinate), higher is the importance of the manifest variable for the latent variable(Dimension).
• The sign of correlation give information about the position of the manifest variable:
  • On the first dimension: Positive sign, the MV is on the right side; Negative sign, the MV is on the left side
  • On the second dimension: Positive sign, the MV is on the upper side; Negative sign, the MV is on the lower side
• We can give a meaning to the two Latent Variables (Dimensions) based on the correlation and the sign.

In our case study, the LVs (dimensions) are so defined:

• \(LV_1\): Right side (cyl, disp, hp, wt); Left side (mpg, drat, vs)
• \(LV_2\). Upper side (am, gear, carb); Lower side (qsec)

round(res.pca1$var$coord[,1:2],3)

      Dim.1  Dim.2
mpg  -0.932  0.026
cyl   0.961  0.071
disp  0.946 -0.080
hp    0.848  0.405
drat -0.756  0.447
wt    0.890 -0.233
qsec -0.515 -0.754
vs   -0.788 -0.377
am   -0.604  0.699
gear -0.532  0.753
carb  0.550  0.673

it is possible to evaluate the correlations between the manifest variables by using the angle between the vectors. For instance, the variables hp, cyl, disp and wt are all positively correlated since all the angles are less then 90°.

plot.PCA(res.pca1, axes = c(1,2), choix = "var")

The quality of the projected manifest variables is given by the Cosine squared.

Below you can find the interpretation intervals:

• 0.00 < \(Cos^2\) <= 0.25: Low Quality
• 0.25 < \(Cos^2\) <= 0.50: Medium-Low Quality
• 0.50 < \(Cos^2\) <= 0.75: Medium-High Quality
• 0.75 < \(Cos^2\) <= 1.00: High Quality

round(res.pca1$var$cos2[,1:2],3)

     Dim.1 Dim.2
mpg  0.869 0.001
cyl  0.924 0.005
disp 0.896 0.006
hp   0.720 0.164
drat 0.572 0.200
wt   0.792 0.054
qsec 0.266 0.569
vs   0.621 0.142
am   0.365 0.489
gear 0.283 0.567
carb 0.303 0.453

7.5.2.3 Units Plot

In the units plot we can evaluate the clusters of units. The clusters define group of units that are similar on the basis of the Latent Variables estimated.

plot.PCA(res.pca1, axes = c(1,2), choix = "ind")

Warning: ggrepel: 10 unlabeled data points (too many overlaps). Consider increasing max.overlaps

7.5.3 PCA with Factoextra Package

library(factoextra)

7.5.3.1 Variables Plot

fviz_pca_var(res.pca1)

7.5.3.2 Units Plot

fviz_pca_ind(res.pca1)

7.5.3.3 BiPlot

Biplot is a plot where both units and variables are represented.

fviz_pca_biplot(res.pca1)

7.5.3.4 Graphical representation of the Eigenvalues

fviz_screeplot(res.pca1)

7.5.3.5 Graphical representation of the Contributions of the manifest variables

fviz_pca_contrib(res.pca1, choice = "var")

Warning in fviz_pca_contrib(res.pca1, choice = "var") :
  The function fviz_pca_contrib() is deprecated. Please use the function fviz_contrib() which can handle outputs  of PCA, CA and MCA functions.